Chapter 7: Introduction to Trigonometry

Trigonometry — literally "triangle measurement" — is one of the oldest and most practical branches of mathematics. Ancient astronomers used it to map the stars, engineers use it to design bridges and buildings, and physicists use it to model waves and oscillations. In this chapter you will learn the foundations of trigonometry: how the ratios of sides in a right triangle give rise to the sine, cosine, and tangent functions; how the unit circle extends these functions to all angles; how radians provide an alternative and often more natural way to measure angles; and how to graph, transform, and invert trigonometric functions. By the end of this chapter you will be prepared for the more advanced identities, equations, and applications that appear in Precalculus and Calculus.

7.1 Right Triangle Trigonometry

Trigonometry begins with right triangles. Given a right triangle with an acute angle $\theta$, we label the three sides relative to that angle: the side directly across from $\theta$ is the opposite side, the side adjacent to $\theta$ (that is not the hypotenuse) is the adjacent side, and the longest side — opposite the right angle — is the hypotenuse.

These ratios depend only on the angle $\theta$, not on the size of the triangle. Any two right triangles that share the same acute angle are similar, so their side ratios are identical. This is the fundamental insight that makes trigonometry so powerful: knowing just one angle and one side of a right triangle is enough to determine all the other sides.

Finding Missing Sides

To find an unknown side of a right triangle, choose the trig ratio that relates the known side, the unknown side, and the given angle. Then solve the resulting equation.

Finding Missing Angles

When you know two sides of a right triangle and need to find an acute angle, compute the appropriate trig ratio and then use the inverse function on your calculator. For instance, if $\sin\theta = 0.6$, then $\theta = \sin^{-1}(0.6) \approx 36.87°$.

Special Right Triangles

Two families of right triangles appear so frequently in mathematics that their side ratios should be memorized. They come from bisecting an equilateral triangle and from cutting a square along its diagonal.

7.2 The Unit Circle

Right triangle trigonometry only covers acute angles ($0° < \theta < 90°$). To extend trigonometric functions to all angles — including obtuse angles, negative angles, and angles greater than $360°$ — we use the unit circle, which is the circle of radius $1$ centered at the origin: $x^2 + y^2 = 1$.

From the equation $x^2 + y^2 = 1$ and the definitions above, we immediately obtain the most important identity in trigonometry:

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the $x$-axis. It always lies between $0°$ and $90°$. To evaluate trig functions at any angle, find its reference angle, look up the trig value of that reference angle, and then apply the correct sign based on the quadrant.

• Quadrant I: reference angle $\alpha = \theta$
• Quadrant II: $\alpha = 180° - \theta$
• Quadrant III: $\alpha = \theta - 180°$
• Quadrant IV: $\alpha = 360° - \theta$

Quadrant Signs: All Students Take Calculus

The signs of sine, cosine, and tangent depend on which quadrant the terminal side falls in. Since $\cos\theta = x$ and $\sin\theta = y$, the signs follow from whether $x$ and $y$ are positive or negative in each quadrant.

The mnemonic "All Students Take Calculus" reads counterclockwise from Quadrant I: All, Sine, Tangent, Cosine. Each word tells you which trig function(s) are positive in that quadrant.

Key Coordinates on the Unit Circle

Combining the special triangle values with the reference angle technique gives us exact coordinates for every multiple of $30°$ and $45°$ around the unit circle. Here are the values in the first quadrant; all other quadrants follow by adjusting signs.

Interactive: The Unit Circle

Drag the slider below to move a point around the unit circle. Watch how the coordinates $(\cos\theta, \sin\theta)$ change as the angle $\theta$ varies.

The point traces the unit circle as the angle slider changes. The dashed lines show the sine and cosine projections.

7.3 Radian Measure

Degrees are intuitive but somewhat arbitrary — why divide a circle into $360$ parts? Radians offer a more natural and mathematically elegant way to measure angles. In calculus and higher mathematics, radians are the standard unit because they simplify many formulas.

Converting Between Degrees and Radians

Since $360° = 2\pi$ radians, we have $180° = \pi$ radians. This gives the conversion factors:

Here are the most commonly encountered degree-radian equivalences:

Arc Length

The relationship between an arc, its central angle, and the radius is beautifully simple when the angle is measured in radians.

This formula is the reason radians are preferred in science and engineering. If we used degrees, the formula would need an extra factor of $\frac{\pi}{180}$, making it less clean.

Sector Area

A sector is the "pie slice" region bounded by two radii and an arc. Its area is a fraction of the full circle's area, proportional to the central angle.

7.4 Graphing Sine and Cosine

The graphs of $y = \sin x$ and $y = \cos x$ are smooth, continuous waves that repeat every $2\pi$ units. These wave patterns model countless real-world phenomena: sound waves, alternating current, tides, pendulum motion, and seasonal temperature cycles. Learning to read and transform these graphs is one of the most important skills in Algebra 2.

The Basic Graphs

The graph of $y = \sin x$ starts at the origin, rises to a peak of $1$ at $x = \frac{\pi}{2}$, returns to $0$ at $x = \pi$, drops to a trough of $-1$ at $x = \frac{3\pi}{2}$, and returns to $0$ at $x = 2\pi$ to complete one full cycle. The graph of $y = \cos x$ has the same shape but starts at its peak: it begins at $1$, drops to $0$ at $\frac{\pi}{2}$, reaches $-1$ at $\pi$, returns to $0$ at $\frac{3\pi}{2}$, and returns to $1$ at $2\pi$.

Key properties shared by both basic graphs:

• Domain: $(-\infty, \infty)$
• Range: $[-1, 1]$
• Period: $2\pi$ (the horizontal length of one complete cycle)
• Amplitude: $1$ (the distance from the midline to a peak or trough)
• Midline: $y = 0$ (the horizontal line halfway between the max and min)

The General Sinusoidal Form

The general form of a transformed sinusoidal function is:

$$y = A\sin\bigl(B(x - C)\bigr) + D$$

(The same form applies with cosine.) Each of the four parameters $A$, $B$, $C$, $D$ controls a specific transformation:

Interactive: Sine/Cosine Transformer

Use the sliders below to adjust $A$, $B$, $C$, and $D$ and watch how the graph of $y = A\sin(B(x - C)) + D$ (blue) transforms compared to the base graph $y = \sin x$ (dashed gray). The dashed green line shows the midline $y = D$.

Drag the sliders to explore amplitude, period, phase shift, and vertical shift in real time.

7.5 Graphing Tangent and Other Trig Functions

The tangent, cotangent, secant, and cosecant functions have graphs that look very different from the smooth waves of sine and cosine. Their defining feature is the presence of vertical asymptotes where the function is undefined.

The Tangent Function

Since $\tan x = \frac{\sin x}{\cos x}$, the function is undefined wherever $\cos x = 0$, which occurs at $x = \frac{\pi}{2} + n\pi$ for any integer $n$. At these values the graph has vertical asymptotes.

For a transformed tangent $y = A\tan(Bx)$, the period becomes $\frac{\pi}{|B|}$ and the vertical stretch factor is $|A|$. Vertical asymptotes shift according to the transformation.

Cotangent, Secant, and Cosecant

The remaining three trig functions are reciprocals of the primary three:

• $\cot x = \frac{\cos x}{\sin x} = \frac{1}{\tan x}$: undefined at $x = n\pi$, period $\pi$, decreasing on each branch.
• $\sec x = \frac{1}{\cos x}$: undefined at $x = \frac{\pi}{2} + n\pi$, period $2\pi$, range $(-\infty, -1] \cup [1, \infty)$.
• $\csc x = \frac{1}{\sin x}$: undefined at $x = n\pi$, period $2\pi$, range $(-\infty, -1] \cup [1, \infty)$.

7.6 Inverse Trigonometric Functions

Trigonometric functions are periodic, so they fail the Horizontal Line Test on their full domains and are not one-to-one. To define inverse functions, we must restrict each trig function to an interval where it is one-to-one. The resulting inverses "undo" the trig functions and are essential for solving trigonometric equations.

Evaluating Inverse Trig Functions

To evaluate an expression like $\arcsin\!\left(\frac{1}{2}\right)$, ask: "What angle $\theta$ in the range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ satisfies $\sin\theta = \frac{1}{2}$?" Since $\sin\frac{\pi}{6} = \frac{1}{2}$ and $\frac{\pi}{6}$ is in the required range, the answer is $\frac{\pi}{6}$ (or equivalently, $30°$).

Compositions of Trig and Inverse Trig Functions

A common type of problem asks you to evaluate a trig function of an inverse trig function, such as $\cos(\arcsin x)$. The strategy is to let $\theta$ equal the inner inverse trig expression, draw a right triangle, label the sides, and then read off the desired trig value.

7.7 Practice Problems

Test your understanding of Chapter 7 with these problems. Click "Show Solution" to reveal a full worked solution for each one.